Strong Stability of Linear Functional Equations with Distributed Delays

This paper considers linear functional equations on $\mathbb R^d$ with distributed delays defined by matrix-valued measures of bounded variation. More precisely, we are interested in providing conditions to ensure that the exponential stability of these systems is preserved under small changes of the parameters which define them. In the special case of difference equations, it is known that exponential stability is preserved under small perturbations of the matrices defining the system, but not of the delays, and an additional condition for preservation of exponential stability under perturbation of the delays is given by the Hale--Silkowski criterion (HSC). In this paper, we extend the treatment of these issues to more general systems. For that purpose, we first put forward an appropriate definition of perturbation on the delays and then propose a conjecture in the spirit of (HSC). We prove several partial results related to that conjecture.